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A Note on Chaotic and Predictable Representations for It\^ O-Markov A NOTE ON CHAOTIC AND PREDICTABLE REPRESENTATIONS FOR ITO-MARKOVˆ ADDITIVE PROCESSES ZBIGNIEW PALMOWSKI, ŁUKASZ STETTNER, AND ANNA SULIMA ABSTRACT. In this paper we provide predictable and chaotic representations for Itˆo-Markov additive processes X. Such a process is governed by a finite-state CTMC J which allows one to modify the pa- rameters of the Itˆo-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itˆo-L´evy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This re- sult generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives. KEYWORDS. Markov additive processes ⋆ martingale representation ⋆ power-jump process ⋆ orthogo- nal polynomials ⋆ stochastic integral ⋆ Brownian motion ⋆ regime switching ⋆ complete market CONTENTS 1. Introduction 2 2. Main results 3 3. Proof of predictable representation 8 3.1. Polynomial representation 8 4. Proofs of main results 13 4.1. Proof of Theorem 2 13 arXiv:1612.09216v2 [math.PR] 25 Aug 2017 4.2. Proof of Theorem 4 15 References 15 1991 Mathematics Subject Classification. 60J30; 60H05. This work was partially supported by the National Science Centre under the grant 2015/17/B/ST1/01102. 1 2 Z. PALMOWSKI —Ł.STETTNER —A.SULIMA 1. INTRODUCTION The Martingale Representation Theorem allows one to represent all martingales, in a certain space, as stochastic integrals. This result is the cornerstone of mathematical finance that produces hedging of some derivatives (see e.g. F¨ollmer and Schied [21]). It is also the basis for the theory of backward stochastic differential equations (see El Karoui et al. [16] for a review). The martingale representation theorem can also lead to Wiener-like chaos expansions, which appear in the Malliavin calculus ( see Di Nunno et al. [15]). The martingale representation theorem is commonly considered in a Brownian setting. It states that any square-integrable martingale M adapted to the natural filtration of the Brownian motion W can be expressed as a stochastic integral with respect to the same Brownian motion W : t (1) M(t)= M(0) + ξ(s)dW (s), Z0 where ξ satisfies suitable integrability conditions. This is the classical Itˆotheorem (see Rogers and Williams [36, Thm. 36.1]). For processes having jumps coming according to a compensated Poisson measure Π¯, the represen- tation theorem was originally established by Kunita and Watanabe in their seminal paper [25, Prop. 5.2] (see also Applebaum [1, Thm. 5.3.5]). In particular, on probability space (Ω, F, P) with augmented natural filtration of a L´evy process {Ht}t≥0, any locally square-integrable {Ht}-martingale M has the following representation: t t (2) M(t)= M(0) + ξ1(s)dW (s)+ ξ2(s, x)Π(d¯ s, dx), R Z0 Z0 Z \{0} where ξ1 and ξ2 are {Ht}- predictable processes satisfying suitable integrability conditions. Later the results were generalized in a series of papers by Chou and Meyer [8], Davis [12] and Elliott [17, 18]. In fact, Elliott [17] was assuming that there are a finite number of jumps within each finite interval. To deal with a general L´evy process (of unbounded variation) Nualart and Schoutens [28] derived a martingale representation using the chaotic representation property in terms of a suitable orthogonal sequence of martingales, obtained as the orthogonalization of the compensated Teugels power jump processes of the L´evy process (see also Corcuera et al. [11]). More on martingale representation can be found in Jacod and Shiryaev [23] , Liptser and Shiryaev [27], Protter [35] and in a review by Davis [13]. The goal of this paper is to derive the martingale representation theorem for an Itˆo-Markov additive process X. Itˆo-Markov additive processes are a natural generalization of Itˆo-L´evy processes and hence also of L´evy processes. The use of Itˆo-Markov additive processes is widespread, making them a classical model in applied probability with a variety of application areas, such as queues, insurance risk, inventories, data communication, finance, environmental problems and so forth (see Asmussen [2], Asmussen and Albrecher [3], Asmussen et al. [4], Asmussen and Kella [5], C¸inlar [9, 10], Prabhu [34, Ch. 7], Pacheco and Prabhu [30], Palmowski and Rolski [33], Palmowski and Ivanovs [32] and references therein). In this paper, we will also derive a chaotic representation of any square-integrable random variable in terms of certain orthogonal martingales. It should be underlined that the derived representation is of predictable type, that is, any square-integrable martingale is a sum of stochastic integrals of some predictable processes with respect to explicit martingales. This representation is hence different from CHAOTIC AND PREDICTABLE REPRESENTATIONS FOR ITOˆ -MARKOV ADDITIVE PROCESSES 3 the one given in (2) (in the jump part) and it can be used to prove completeness of some financial markets. This paper is organized as follows. In Section 2 we present the main results. Their proofs are given in Section 4, preceeded by a crucial preliminary fact presented in Section 3. 2. MAIN RESULTS We begin by defining Itˆo-Markov additive processes. Let (Ω, F, P) be a complete probability space and let T := [0; T ] be a finite time horizon, where 0 <T < ∞ is fixed. On this probability space we consider a homogeneous continuous-time Markov chain J := {J(t): t ∈ T} with a finite state space. For simplicity, we follow the notation of Elliott et al. [19] and we identify the state space as a N finite set of unit vectors E := {e1,..., eN } of R . Here the jth component of ei is the Kronecker delta δij for each i, j = 1, 2,...,N. Moreover, the Markov chain J is characterized by an intensity matrix [λij ]i,j=1,2,...,N . The element λij is the transition intensity of the Markov chain J jumping from state ei to state ej . A process (J, X)= {(J(t),X(t)) : t ∈ T} on the state space {e1,..., eN }×R is a Markov additive process (MAP) if (J, X) is a Markov process and the conditional distribution of (J(s + t),X(s + t) − X(s)) for s,t ∈ T, given (J(s),X(s)), depends only on J(s) (see C¸inlar [9, 10]). Every MAP has a very special structure. It is usually said that X is the additive component and J is the background process representing the environment. Moreover, the process X evolves as a L´evy process while J(t) = ej . Following Asmussen and Kella [5] we can decompose the process X as follows: (3) X(t)= X(t)+ X(t), where N (4) X(t) := Ψi(t) i=1 X for (i) (5) Ψi(t) := Un 1{J(Tn)=ei, Tn≤t} n X≥1 (i) and for the jump epochs {Tn} of J. Here Un (n ≥ 1, 1 ≤ i ≤ N) are independent random variables (i) which are also independent of X such that for every i, the random variables Un are identically distributed. Note that we can express the process Ψi as follows: t i Ψi(t)= x ΠU (ds, dx) R Z0 Z for the point measures i P (i) (6) ΠU ([0,t], dx) := (Un ∈ dx)1{J(Tn)=ei, Tn≤t}, i =1, . , N. n X≥1 (ij) Remark 1. One can consider jumps U with distribution depending also on the state ej the Markov chain is jumping to by extending the state space to the pairs (ei, ej ) (see Gautam et al. [22, Thm. 5] for details). The first component in equation (3) is an Itˆo-L´evy process and it has the following decomposition (see Oksendal and Sulem [29, p. 5]): t t t (7) X(t) := X(0) + µ0(s)ds + σ0(s)dW (s)+ γ(s−, x)Π(d¯ s, dx), R Z0 Z0 Z0 Z 4 Z. PALMOWSKI —Ł.STETTNER —A.SULIMA where W denotes the standard Brownian motion independent of J, Π(d¯ t, dx) := Π(dt, dx) − ν(dx)dt is the compensated Poisson random measure which is independent of J and W and N i (8) µ0(t) := hµ0,J(t)i = µ0hei,J(t)i, i=1 X N i (9) σ0(t) := hσ0,J(t)i = σ0hei,J(t)i, i=1 X N (10) γ(t, x) := hγ(x),J(t)i = γi(x)hei,J(t)i, i=1 X 1 N ′ RN 1 N ′ RN for some vectors µ0 = (µ0,...,µ0 ) ∈ , σ0 = (σ0 ,...,σ0 ) ∈ and the vector-valued measur- able function γ(x) = (γ1(x),...,γN (x)). The measure ν is the so-called jump-measure identifying the distribution of the sizes of the jumps of the Poisson measure Π. The components X and X in (3) are, conditionally on the state of the Markov chain J, independent. Additionally, we suppose that the L´evy measure satisfies, for some ε> 0 and λ> 0, (11) exp(λ|γ(s−, x)|)ν(dx) < ∞, P − a.e., exp(λx)P(U (i) ∈ dx) < ∞ c c Z(−ε,ε) Z(−ε,ε) for i = 1,...,N, where U (i) is a generic size of jump when the Markov chain J is jumping into state i.
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